Vectors Vectors and Scalars Properties of Vectors Components of a - - PDF document

Vectors

• Vectors and Scalars
• Properties of Vectors
• Components of a Vector and Unit Vectors
• Homework

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Vectors and Scalars

• Vector - quantity that has magnitude and direction

– e.g. displacement, velocity, acceleration, force

• Scalar - quantity that has only magnitude

– e.g. Time, mass, energy 2

Displacement Vector

As a particle moves from A to B along the path repre- sented by the dashed curve, its displacement is the vector shown by the arrow from A to B.

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Adding Vectors

When vectorB is added to vectorA, the resultant Ris the vector that runs from the tail ofAto the head ofB.

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Commutative Property of Vector Addition

• The vectorRresulting from the addition of the vectors

A and B is the diagonal of a parallelogram of sides A andB.

• Vector addition is commutative, that isA+B=B+A.

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Associative Property of Vector Addition

A+(B+C) = (A+B)+C

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Subtraction of Vectors

• To subtract vector B from vector A, simply add the

vector -Bto vectorA.

• The vector -B is equal in magnitude and opposite in

direction to the vectorB.

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Components of a Vector

A vectorAlying in the xy plane can be represented by its component vectorsAx andAy. Ax = A cos θ Ay = A sin θ A =

• A2

x + A2 y

tan θ = Ay Ax

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Unit Vectors

• The unit vectors i, j, and k are directed along the x, y,

and z axes, respectively.

• The unit vectors i,j, and k form a set of mutually per-

pendicular vectors and the magnitude of each unit vector is one

– |i|=|j|=|k|=1 9

Vectors in Component Form

A vector A lying in the xy plane has component vectors Axiand Ayjwhere Ax and Ay are the components ofA. A=Axi+ Ayj

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Example 1

A small plane leaves an airport on an overcast day and later is sighted 215 km away, in a direction making an angle of 22◦ east of north. (a) How far east and north is the airplane from the airport when sighted? (b) Using a coordinate system with the y-axis pointing north and the x-axis east, write the position of the airplane in unit vector notation.

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Example 1 Solution

A small plane leaves an airport on an overcast day and later is sighted 215 km away, in a direction making an angle of 22◦ east of north. (a) How far east and north is the airplane from the airport when sighted?

rx ry y x r θ N

θ = 90◦ − 22◦ = 68◦ rx = r cos θ = (215 km) cos 68◦ = 81 km ry = r sin θ = (215 km) sin 68◦ = 199 km (b) Using a coordinate system with the y-axis pointing north and the x-axis east, write the position of the air- plane in unit vector notation. r = rxi + ryj = (81 km) i + (199 km) j

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Vector Addition Using Components

R = A + B Rxi + Ryj = (Axi + Ayj) + (Bxi + Byj) Rxi + Ryj = (Ax + Bx) i + (Ay + By) j Rx = Ax + Bx Ry = Ay + By

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Example 2

Find R = A + B + C where A = 4.2i - 1.6j, B = -3.6i + 2.9j, andC= -3.7j.

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Example 2 Solution

Find R = A + B + C where A = 4.2i - 1.6j, B = -3.6i + 2.9j, andC= -3.7j. R = Rxi + Ryj R = (Ax + Bx + Cx) i + (Ay + By + Cy) j R = (4.2 − 3.6 + 0) i + (−1.6 + 2.9 − 3.7) j R = 0.6i − 2.4j

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Homework Set 5 - Due Mon. Sept. 20

• Read Sections 1.8-1.10
• Do Problems 1.35, 1.44, 1.52 & 1.53

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